The Intermediate Value Theorem is one of the most important theorems in Introductory Calculus, and it forms the basis for proofs of many results in subsequent and advanced Mathematics courses. The
** history ** of this theorem begins in the 1500's and is eventually based on the academic work of Mathematicians Bernard Bolzano, Augustin-Louis Cauchy, Joseph-Louis Lagrange, and Simon Stevin. Generally speaking, the Intermediate Value Theorem applies to continuous functions and is used to prove that equations, both ** alebraic ** and ** transcendental ** , are solvable. The formal statement of this theorem together with an illustration of the theorem follow.

INTERMEDIATE VALUE THEOREM: Let $f$ be a continuous function on the closed interval $ [a, b] $. Assume that $m$ is a number ($y$-value) between $f(a)$ and $f(b)$. Then there is at least one number $c$ ($x$-value) in the interval $[a, b]$ which satifies
$$ f(c)=m $$

While the result certainly seems intuitively obvious, the formal proof of the Intermediate Value Theorem is quite sophisticated and is beyond the experience of most first-year calculus students. For a simple illustration of the this theorem, assume that a function $f$ is a continuous and $m=0$. Then the conditions $ f(a)<0 $ and $ f(b)>0 $ would lead to the conclusion that the equation $f(x)=0$ is solvable for $x$, i.e., $f(c)=0$. This result is formally accredited to Berard Bolzano and is called

Here is a summary of how I will use the Intermediate Value Theorem in the problems that follow.

$ \ \ \ \ $ 1. Define a function $ y=f(x)$.

$ \ \ \ \ $ 2. Define a number ($y$-value) $m$.

$ \ \ \ \ $ 3. Establish that $f$ is continuous.

$ \ \ \ \ $ 4. Choose an interval $ [a, b] $.

$ \ \ \ \ $ 5. Establish that $m$ is between $ f(a) $ and $ f(b) $.

$ \ \ \ \ $ 6. Now invoke the conclusion of the Intermediate Value Theorem.

In the list of Differentials Problems which follows, most problems are average and a few are somewhat challenging.

- PROBLEM 1 : Use the Intermediate Value Theorem to prove that the equation $ x^3+2x-5=0 $ is solvable.

Click HERE to see a detailed solution to problem 1.

- PROBLEM 2 : Use the Intermediate Value Theorem to prove that the equation $ e^x+x+2=0 $ is solvable.

Click HERE to see a detailed solution to problem 2.

- PROBLEM 3 : Use the Intermediate Value Theorem to prove that the equation $ x^5-3x=4-x^2 $ is solvable.

Click HERE to see a detailed solution to problem 3.

- PROBLEM 4 : Use the Intermediate Value Theorem to prove that the equation $ x^3=20+ \sqrt{x} $ is solvable.

Click HERE to see a detailed solution to problem 4.

Click HERE to return to the original list of various types of calculus problems.

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Duane Kouba ...
October 24, 2019